
theorem Th22:
  for N being invertible Matrix of 3,F_Real
  for h being Element of SubGroupK-isometry
  for n11,n12,n13,n21,n22,n23,n31,n32,n33 being Element of F_Real
  for P being Element of absolute
  for u being non zero Element of TOP-REAL 3 st
  h = homography(N) & N = <* <* n11,n12,n13 *>,
                             <* n21,n22,n23 *>,
                             <* n31,n32,n33 *> *> &
  P = Dir u & u.3 = 1 holds n31 * u.1 + n32 * u.2 + n33 <> 0
  proof
    let N be invertible Matrix of 3,F_Real;
    let h be Element of SubGroupK-isometry;
    let n11,n12,n13,n21,n22,n23,n31,n32,n33 be Element of F_Real;
    let P be Element of absolute;
    let u be non zero Element of TOP-REAL 3;
    assume
A1: h = homography(N) & N = <* <* n11,n12,n13 *>,
                               <* n21,n22,n23 *>,
                               <* n31,n32,n33 *> *> &
    P = Dir u & u.3 = 1;
    reconsider Q = homography(N).P as Point of ProjectiveSpace TOP-REAL 3;
    reconsider Q = homography(N).P as Element of absolute by A1,BKMODEL3:35;
    ex v be non zero Element of TOP-REAL 3 st Q = Dir v & v.3 = 1 &
      absolute_to_REAL2 Q = |[v.1,v.2]| by BKMODEL1:def 8;
    hence thesis by A1,Th21;
  end;
